Low Thrust Minimum-Fuel Orbital Transfer: A Homotopic Approach

Transcription

1 Low Thrust Minimum-Fuel Orbital Transfer: A Homotopic Approach 3rd International Workshop on Astrodynamics Tools and Techniques ESA, DLR, CNES ESTEC, Noordwijk Joseph Gergaud and Thomas Haberkorn 2 5 October 26 / 23

2 Orbital Transfer Problem Modelisation Homotopy Method Shooting method Homotopy PC methods Numerical results Convergence of the method Local minima Examples of solutions Links with impulsive case Conclusion Conclusion Bibliography 2/ 23

3 Modelisation Orbital transfer ORBITE FINALE r 2 4 2!2 r 3 5!5 4 2 r 2!2!4!4!5 5 r!4 r 3 ORBITE INITIALE 5!2 r 2 4!5!5 5 r 2 LEO GOE Transfer: P =.625 Mm, e =.75 and i = 7 Initial Mass: m = 5 kg Low Thrust: T max =. N Supported by 3/ 23

4 Modelisation Coordinates equatorial plan X J2 neglected Earth gravitational force r = µ r 3 r + T m Modified Gauss Coordinates x = (P, e x, e y, h x, h y, L): orbit! Z satellite v " Y i perigee P = a( e 2 ) e x = e cos(ω + ω) e y = e sin(ω + ω) h x = tan(i/2) cos Ω h y = tan(i/2) sin Ω L = Ω + ω + ν + 2πn (cumulative) n = Number of revolutions 4/ 23

5 Modelisation Control k O j i r w S v s q Transverse reference frame (q, s, w) We normalize the control T = T max u So the constraint ( on the control ) is u u = u 2 + u2 2 + u2 3 State equation: ẋ = f (x) + T max /m 3 i= f i(x)u i ṁ = T max I sp g u 5/ 23

6 Modelisation Cost Preliminary results min t f t f min T max c te (T. Le, S. Geffroy, R. Épenoy) min L f (L f min L )T max c te (T. Haberkorn) Maximization of the final mass: max m(t f ) Minimization of the consumption: min t f u dt 6/ 23

7 Modelisation Optimal Control Problem (P) min t f u dt ẋ = f (x) + T max /m 3 i= u if i (x) ṁ = βt max u u Initial and Final Conditions L f free t f fixed We define the constant c t f > such that t f = c t f t f min 7/ 23

8 Shooting method Difficulties of the shooting method The optimal control is discontinuous: The engine is off or on with the maximal thrust We don t know the number of switching times and their locations Difficulties to find an initial guess 8/ 23

9 Homotopy Main idea Cost J λ (u) = t f ( λ) u 2 + λ u dt or t f Problem easy to solve for λ = Regular problem for λ < Original problem for λ = u (2 λ) dt Optimal Control Problems (P λ ) Boundary Value Problems (BVP λ ) Shooting Homotopy S(z, λ) 9/ 23

10 PC methods Homotopy algorithms Path following of the zeros curve of the homotopy function: S : R n [, ] R n (z, λ) S(z, λ) Global Newton algorithm Initialisation z solution of S(z, ) = = λ < λ <... < λ n = for i =,..., n do solve S(z, λ i ) = by Newton method with initial point z i but for our orbital transfer problem we diverge How to choose the sequence (λ i ) i? = Homotopy methods (Allgower and Georg) Predictor-Corrector methods Piecewise Linear methods / 23

11 PC methods Predictor Corrector methods Tangent vector for the prediction S(z, λ) = lambda.6.5 (ez, e λ).4.3 Prediction.2 (z λ, λ).!8!6!4! z / 23

12 PC methods Predictor Corrector methods Tangent vector for the prediction. Come back on the zeros path (correction). lambda S(z, λ) = (z λ +, λ + ) Correction (ez, λ) e Prediction.2 (z λ, λ).!8!6!4! z / 23

13 PC methods Predictor Corrector methods Tangent vector for the prediction. Come back on the zeros path (correction). Until λ =. lambda S(z, λ) = (z λ +, λ + ) Correction (ez, λ) e Prediction.2 (z λ, λ).!8!6!4! z / 23

14 PC methods MfMax software We used the software HOMPACK9 (Watson and al.) for PC-methods We compute S (c(s)) by finite differences and S(c(s)) by numerical integration (rkf45) = we must have a good adequation between the step of finite differences and the local errors in numerical integration in order to have a good approximation of the derivative S (c(s)) 2/ 23

15 Convergence of the method Path following.5 (!! ) u 2 +! u !.5!!.5.5!3.2!3!2.8!2.6! u 2!! !.5!!.5.5!3.2!3!2.8!2.6!2.4.5 Smooth path ! Discret Differentielle z S(z,!) = Advantage of PC methods 3/ 23

16 Convergence of the method Control with respect to λ T max = N Critere convexe! =,.5 et.8.6 u t Critere puissance! =,.5 et.8.6 u t 4/ 23

17 Local minima t f fixed and L f free: local minima 39 Lots of minima because the number of revolutions is free m f t f 5/ 23

18 Local minima t f fixed and L f free: local minima 39 Lots of minima because the number of revolutions is free We fixe L f : L f = c L f (L f min L ) + L t f free m f t f 5/ 23

19 Local minima m f vs c L f 395 t L m f does not depend on T max (for c L f fixed) Tmax = N Tmax = N Tmax =.5 N Tmax =. N Impulse L f = c L f (L f min L ) + L Limit mass = mass of the impulsive case / 23

20 Examples of solutions Trajectory for T max = N et c L f = 2 r 3 5!5 Thrust arcs on all apogees. Thrust arcs on last perigees. r r 2!2!4!4 r 3 2!2 r 2 4!2!!4!6!4!2 2 4 r!2!4!2 2 4 r 2 7/ 23

21 Examples of solutions Other initial orbits (P, e ) = (.625,.5) (P, e ) = (2,.75) r 2 r 2!!!2!2!3!3!4!4!5!4!3!2! r!5!8!6!4! / 23

22 Examples of solutions Trajectory for T max =. N r 3 5! ! r 2!2!3!4!4!2 r More than 75 revolutions More than 5 switching times 9/ 23

23 Links with impulsive case Links with impulsive case In the coplanar transfer case, if T max + and [L, L f ] contains exactly one apogee and perigee then the control converges to the impulsive solution When c L f then m f final mass in the impulsive case Tmax = N Tmax = N Tmax =.5 N Tmax =. N Impulse / 23

24 Links with impulsive case Links with impulsive case q! t s! t w! t u t Thrust in time for T max = N, c L f = 5 and i = 5 There are three strategies in time for the optimal control: thrust at perigees, thrust at apogee and thrust at perigee, as in the impulsive case. 2/ 23

25 Conclusion Conclusion and developments Conclusion: We have completely solve the problem without any knowledge on the solution (number and location of switching times,...) Developments Interplanetary transfer? State constraints? 22/ 23

26 Bibliography Bibliography J. Gergaud et T. Haberkorn. Homotopy method for minimum consumption orbit transfer problem. Control, Optimisation and Calculus of Variations, Vol. 2(2)294:3, April 26 J. Gergaud, T. Haberkorn and P. Martinon. Low thrust minimum-fuel orbital transfer: a homotopic approach, Journal of Guidance, Control, and Dynamics, Vol. 27(6)46:6, Nov. 24 T. Haberkorn. Transfert orbital à poussée faible avec minimisation de la consommation: résolution par homotopie différentielle, PhD ENSEEIHT IRIT, 8 octobre 24 J. Gergaud, T. Haberkorn and J. Noailles. MfMax(v & v): Method explanation manual, Technical report ENSEEIHT-IRIT, UMR CNRS 555, RT/APO/4/, january 24, 23/ 23